Skip to main content

Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups


We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger–Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.

This is a preview of subscription content, access via your institution.


  1. 1.

    Abbena, E., Grassi, A.: Hermitian left invariant metrics on complex lie groups and cosymplectic Hermitian manifolds. Boll. Un. Mat. Ital. A 6, 371–379 (1986)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Arroyo, R., Lafuente, R.: The long-time behavior of the homogeneous pluriclosed flow. Proc. London Math. Soc. 119(1), 266–289 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Böhm, C. Lafuente, R.: Real geometric invariant theory. arXiv e-prints, (2017). arXiv:1701.00643

  4. 4.

    Enrietti, N., Fino, A., Vezzoni, L.: The pluriclosed flow on nilmanifolds and tamed symplectic forms. J. Geom. Anal. 25(2), 883–909 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Fei,T., Phong, D.: Unification of the Kähler-Ricci and Anomaly flows. arXiv e-prints, (2019). arXiv:1905.02274

  6. 6.

    Lafuente, R., Pujia, M., Vezzoni, L.: Hermitian curvature flow on unimodular Lie groups and static invariant metrics. Trans. Am. Math. Soc. 373(6), 3967–3993 (2020)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Lauret, J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Lauret, J.: Convergence of homogeneous manifolds. J. Lond. Math. Soc. 86(3), 701–727 (2012)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lauret, J.: Ricci flow of homogeneous manifolds. Math. Z. 274, 373–403 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lauret, J.: Curvature flows for almost-Hermitian Lie groups. Trans. Am. Math. Soc. 367, 7453–7480 (2015)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lauret, J.: Geometric flows and their solitons on homogeneous spaces. Rend. Semin. Mat. Univ. Politec. Torino 74(1), 55–93 (2016)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Panelli, F., Podestà,F.: Hermitian Curvature Flow on compact homogeneous spaces. J. Geom. Anal. (2019).

  13. 13.

    Pediconi, F., Pujia, M.: Hermitian curvature flow on complex locally homogeneous surfaces. Ann. Mat. Pura Appl. (2020).

  14. 14.

    Phong, D., Picard, S., Zhang, X.: Anomaly flows. Comm. Anal. Geom. 26(4), 955–1008 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Phong, D., Picard, S., Zhang, X.: Geometric flows and Strominger systems. Math. Z. 288(1–2), 101–113 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Phong, D., Picard, S., Zhang, X.: The Anomaly flow and the Fu-Yau equation. Ann. PDE 4, 13 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Phong, D., Picard, S., Zhang, X.: A flow of conformally balanced metrics with Kähler fixed points. Math. Ann. 374(3–4), 2005–2040 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Phong, D., Picard, S., Zhang, X.: The Anomaly flow on unimodular Lie groups. Contemp. Math. 735, 217–237 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Pujia, M.: Expanding solitons to the Hermitian curvature flow on complex Lie groups. Differ. Geom. Appl. 64, 201–216 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Pujia, M., Ugarte, L.: The Anomaly flow on nilmanifolds. arXiv e-prints, (2020). arXiv:2004.06744

  21. 21.

    Pujia, M., Vezzoni, L.: A remark on the Bismut–Ricci form on 2-step nilmanifolds. C. R. Acad. Sci. Paris, Ser. I 356, 222–226 (2018)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Stanfield, J.: The Hermitian Curvature Flow on almost-abelian complex Lie groups. In preparation, (2020)

  23. 23.

    Streets, J.: Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifold. Comm. Partial Differ. Equ. 41(2), 318–374 (2016)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Streets, J.: Pluriclosed flow on generalized Kähler manifolds with split tangent bundle. J. Reine Angew. Math. 739, 241–276 (2018)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 16, 3101–3133 (2010)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Streets,J., Tian, G.: Hermitian curvature flow. J. Eur. Math. Soc, 13(601-634), (2011)

  27. 27.

    Streets, J., Tian, G.: Generalized Kähler geometry and the pluriclosed flow. Nuclear Phys. B 858(2), 366–376 (2012)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Streets, J., Tian, G.: Regularity results for pluriclosed flow. Geom. Top. 17(4), 2389–2429 (2013)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Ustinovskiy, Y.: Hermitian curvature flow on complex homogeneous manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2019),

  30. 30.

    Ustinovskiy, Y.: The Hermitian curvature flow on manifolds with non-negative Griffiths curvature. Am. J. Math. 141(6), 1751–1775 (2019)

    MathSciNet  Article  Google Scholar 

Download references


The author warmly thanks Luigi Vezzoni and Ramiro Lafuente for their interest and helpful comments.

Author information



Corresponding author

Correspondence to Mattia Pujia.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pujia, M. Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups. manuscripta math. 166, 237–249 (2021).

Download citation

Mathematics Subject Classification

  • Primary 53C44
  • Secondary 53C15
  • 53C07
  • 53B15